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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Final Year)
MATHEMATICS & STATISTICS
Math 325/425 : Representation Theory of Finite Groups 2 hours
You should answer ALL questions in Section A and TWO questions in Section B.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is 40
SECTION A
A1. (a) Let^ R^ be a ring, let^ M, N^ be two^ R-modules and let^ ฯ^ :^ M^ โ^ N^ be an^ R-homomorphism. (i) Define the kernel, ker(ฯ), and image, im(ฯ), of ฯ. [2] (ii) Prove that im(ฯ) is an R-submodule of N. [4] (b) Let R = M 2 (R) be the ring of 2 ร 2 real matrices and consider R as an R-module in the natural way. Let ฯ : R โ R be defined by
ฯ
a b c d
a 0 c 0
Show that ฯ is an R-homomorphism and find its kernel. [4]
A2. Let G = C 6 = ใaใ, and V = ใv 1 , v 2 ใ be the CG-module defined by av 1 = 3v 1 โ v 2 , av 2 = 7v 1 โ 2 v 2. Find all CG-submodules of G. [10] A3. Let G = D 8 = { 1 , a, a^2 , a^3 , b, ba, ba^2 , ba^3 }, where a^4 = 1 = b^2 and ab = baโ^1 ; set
A =
, B =
Show that the map ฯ : G โ GL 2 (C) defined by ฯ(biaj^ ) = BiAj^ for 0 โค i โค 1 , 0 โค j โค 3 is a representation of G. [10] please turn over
SECTION A continued
A4. (^) (a) Let G be a subgroup of the symmetric group Sn. Define the permutation module V for G. [2] (b) Let G = S 3 , and let V = ใv 1 , v 2 , v 3 ใ be the permutation module. Let B 1 be the standard basis v 1 , v 2 , v 3 , and B 2 be the basis v 1 + v 2 + v 3 , v 1 โ v 2 , v 2 โ v 3. Calculate the matrices [g]B 1 and [g]B 2 as g runs through G. [8]
A5. Let G = C 4 = ใaใ, and V = ใv 1 , v 2 ใ and W = ใw 1 , w 2 ใ be the CG-modules defined by
av 1 = 2v 1 โ v 2 , aw 1 = โ 3 w 1 + 10w 2 , av 2 = 5v 1 โ 2 v 2 , aw 2 = โw 1 + 3w 2.
Find a basis for the vector space HomCG(V, W ). [10]
please turn over
SECTION B continued
B3. The following is the Cayley table of a non-abelian group G of order 8 with generators a and b. G e a a^2 a^3 b ba ba^2 ba^3 e e a a^2 a^3 b ba ba^2 ba^3 a a a^2 a^3 e ba^3 b ba ba^2 a^2 a^2 a^3 e a ba^2 ba^3 b ba a^3 a^3 e a a^2 ba ba^2 ba^3 b b b ba ba^2 ba^3 a^2 a^3 e a ba ba ba^2 ba^3 b a a^2 a^3 e ba^2 ba^2 ba^3 b ba e a a^2 a^3 ba^3 ba^3 b ba ba^2 a^3 e a a^2 Define elements v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8 โ CG by v 1 = e + a^2 , v 2 = a + a^3 , v 3 = b + ba^2 , v 4 = ba + ba^3 , v 5 = e โ a^2 , v 6 = a โ a^3 , v 7 = b โ ba^2 , v 8 = ba โ ba^3. (a) Set U = ใv 1 , v 2 , v 3 , v 4 ใ, W = ใv 5 , v 6 , v 7 , v 8 ใ. By considering the elements avk and bvk for 1 โค k โค 8, show that U and W are CG- submodules of CG, and that CG = U โ W. [10] (b) By considering elements of the form v 1 ยฑ v 2 ยฑ v 3 ยฑ v 4 for appropriate choices of signs, find four 1-dimensional CG-submodules U 1 , U 2 , U 3 and U 4 of U , and show that U = U 1 โ U 2 โ U 3 โ U 4. [10] (c) Write w 1 = v 5 + iv 6 , w 1 โฒ^ = v 7 + iv 8 , w 2 = v 5 โ iv 6 , w 2 โฒ^ = v 7 โ iv 8 , and set Wk = ใwk, wkโฒใ for k = 1, 2. Show that W 1 and W 2 are two 2-dimensional CG-submodules of W , and that W = W 1 โ W 2. [10]
end of exam
